The Classic Greek Ladder and Newton's Method

Introduction

For many students in early mathematics courses, their familiarity with approximations is limited to \( \sqrt{2}\approx{1.414} \), \( \sqrt{3}\approx{1.732} \), \( \pi\approx{\frac{22}{7}} \), and maybe a few more. But a topic of number theory, Diophantine Approximations (honoring Diophantus, a mathematician of Alexandria who lived circa 207 - 291 AD and wrote books called Arithmetica), involves approximating irrational numbers by ordinary reduced fractions. One of the approximation "tools" of ancient mathematicians is a construct called Greek ladders. Maybe Greek ladders will ignite your interest in approximations by ordinary fractions.

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